Phase Transition is associated with a drastic change in some observable (ordered parameter) of the system when the controlled parameter is tuned smoothly. Lee-Yang theory of phase transition is discussed which is related to the accumulation of singularities of free energy, equivalently complex roots of Grand Partition function (Partition function) at points on positive real axis in complex fugacity plane; and more general (p+1) phase system is discussed, and also the case when (w+2) phases coexist together. Comparison to Mayer's theory is also presented.
We show that the generating functions of avalanche observables in SOC models exhibits a Lee-Yang phenomenon. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeroes is proposed including finite sampling effects.
The densities of Yang-Lee zeros for the Ising ferromagnet on the $L\times L$ square lattice are evaluated from the exact grand partition functions ($L=3\sim16$). The properties of the density of Yang-Lee zeros are discussed as a function of temperature $T$ and system size $L$. The three different classes of phase transitions for the Ising ferromagnet, first-order phase transition, second-order phase transition, and Yang-Lee edge singularity, are clearly distinguished by estimating the magnetic scaling exponent $y_h$ from the densities of zeros for finite-size systems. The divergence of the den…
The distributions of the Yang-Lee zeros of the ferromagnetic and antiferromagnetic Q-state Potts models in one dimension are studied for arbitrary Q and temperature. The Yang-Lee zeros of the Potts antiferromagnet have been fully investigated for the first time. The distributions of the Yang-Lee zeros show a variety of different shapes. Some of the Yang-Lee zeros lie on the positive real axis even for T>0. For the ferromagnetic model this happens only for Q<1, while there exist some zeros of the antiferromagnetic model on the positive real axis both for Q<1 and for Q>1.
This paper studies the Yang-Lee edge singularity of 2-dimensional (2D) Ising model based on a quantum spin chain and transfer matrix measurements on the cylinder. Based on finite-size scaling, the low-lying excitation spectrum is found at the Yang-Lee edge singularity. Based on transfer matrix techniques, the single structure constant is evaluated at the Yang-Lee edge singularity. The results of both types of measurements are found to be fully consistent with the predictions for the (A4;A1)minimal conformal field theory, which was previously identified with this critical point.
This paper studies the Yang-Lee edge singularity of 2-dimensional 2D Ising model through a quantum spin chain. In particular, finite-size scaling measurements on the quantum spin chain are used to determine the low-lying excitation spectrum and central charge at the Yang-Lee edge singularity. The measured values are consistent with predictions for the (A4, A1) minimal conformal field theory.
We present a pedagogical account of the Lee-Yang theory of equilibrium phase transitions and review recent advances in applying this theory to nonequilibrium systems. Through both general considerations and explicit studies of specific models, we show that the Lee-Yang approach can be used to locate and classify phase transitions in nonequilibrium steady states.
The Yang-Lee zeros of the Q-state Potts model on recursive lattices are studied for non-integer values of Q. Considering 1D lattice as a Bethe lattice with coordination number equal to two, the location of Yang-Lee zeros of 1D ferromagnetic and antiferromagnetic Potts models is completely analyzed in terms of neutral periodical points. Three different regimes for Yang-Lee zeros are found for Q>1 and 0<Q<1. An exact analytical formula for the equation of phase transition points is derived for the 1D case. It is shown that Yang-Lee zeros of the Q-state Potts model on a Bethe lattice are located …
The density of zeros of the partition function of the Ising model on a class of treelike lattices is studied. An exact closed-form expression for the pertinent critical exponents is derived by using a couple of recursion relations which have a singular behavior near the Yang-Lee edge.
Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang-Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions.
A general analytical formula for recurrence relations of multisite interaction Ising models in an external magnetic field on the Cayley-type lattices is derived. Using the theory of complex analytical dynamics on the Riemann sphere, a numerical algorithm to obtain Yang-Lee and Fisher zeros of the models is developed. It is shown that the sets of Yang-Lee and Fisher zeros are almost always fractals, that could be associated with Mandelbrot-like sets on the complex magnetic field and temperature planes respectively.
We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibri…
In order to provide an experimental access to the statistical theory of Lee and Yang (1952) the density function g(theta) of zeros on the Lee-Yang circle has been determined for the first time by analyzing isothermal magnetization data, m(H), of the Ising ferromagnet FeCl2 in axial magnetic fields, H, at temperatures 34 < T < 99K. The validity of our approach is demonstrated by the perfect agreement of magnetic specific heat data as calculated from g(theta) and m(H) via Maxwells relation, respectively. Moreover, the correct in-plane exchange constant of FeCl2 emerges from the series expansion …
We apply the Yang-Lee theory of phase transitions to an urn model of separation of sand. The effective partition function of this nonequilibrium system can be expressed as a polynomial of the size-dependent effective fugacity $z$. Numerical calculations show that in the thermodynamic limit, the zeros of the effective partition function are located on the unit circle in the complex $z$-plane. In the complex plane of the actual control parameter certain roots converge to the transition point of the model. Thus the Yang-Lee theory can be applied to a wider class of nonequilibrium systems than tho…
We consider how the Lee-Yang description of phase transitions in terms of partition function zeros applies to nonequilibrium systems. Here one does not have a partition function, instead we consider the zeros of a steady-state normalization factor in the complex plane of the transition rates. We obtain the exact distribution of zeros in the thermodynamic limit for a specific model, the boundary-driven asymmetric simple exclusion process. We show that the distributions of zeros at the first and second order nonequilibrium phase transitions of this model follow the patterns known in the Lee-Yang…
Using both the exact enumeration method (microcanonical transfer matrix) for a small system (L = 9) and the Wang-Landau Monte Carlo algorithm for large systems to L = 30, we obtain the exact and approximate densities of states g(M,E), as a function of magnetization M and exchange energy E, for the triangular-lattice Ising model. Based on the density of states g(M,E), we investigate the phase transition properties of Yang-Lee zeros for the triangular Ising antiferromagnets and obtain the magnetic exponents at various temperatures.
We show here for the one-dimensional spin-1/2 ANNNI (axial-next-to-nearest-neighbor-Ising) model in an external magnetic field that the linear density of Yang-Lee zeros may diverge with critical exponent $σ= -2/3$ at the Yang-Lee edge singularity. The necessary condition for this unusual behavior is the triple degeneracy of the transfer matrix eigenvalues. If this condition is absent we have the usual value $σ= -1/2$. Analogous results have been found in the literature in the spin-1 Blume-Emery-Griffths model and in the three-state Potts model in a magnetic field with two complex components. O…
Lattice and continuum fluid models with repulsive-core interactions typically display a dominant, critical-type singularity on the real, negative activity axis. Lai and Fisher recently suggested, mainly on numerical grounds, that this repulsive-core singularity is universal and in the same class as the Yang-Lee edge singularities, which arise above criticality at complex activities with positive real part. A general analytic demonstration of this identification is presented here using a field-theory approach with separate representation of the repulsive and attractive parts of the pair interac…
The microcanonical transfer matrix is used to study the distribution of Yang-Lee zeros of the $Q$-state Potts model in the complex magnetic-field ($x=e^{βh}$) plane for the first time. Finite size scaling suggests that at (and below) the critical temperature the zeros lie close to, but not on, the unit circle with the two exceptions of the critical point $x=1$ ($h=0$) itself and the zeros in the limit T=0.
First order phase transitions in finite systems can be defined through the bimodality of the distribution of the order parameter. This definition is equivalent to the one based on the inverted curvature of the thermodynamic potential. Moreover we show that it is in a one to one correspondence with the Yang Lee theorem in the thermodynamic limit. Bimodality is a necessary and sufficient condition for zeroes of the partition sum in the control intensive variable complex plane to be distributed on a line perpendicular to the real axis with a uniform density, scaling like the number of particles.
Equilibrium systems which exhibit a phase transition can be studied by investigating the complex zeros of the partition function. This method, pioneered by Yang and Lee, has been widely used in equilibrium statistical physics. We show that an analogous treatment is possible for a nonequilibrium phase transition into an absorbing state. By investigating the complex zeros of the survival probability of directed percolation processes we demonstrate that the zeros provide information about universal properties. Moreover we identify certain non-trivial points where the survival probability for bond…
It is known that the (exact) renormalization transformations for the one-dimensional Ising model in field can be cast in the form of a logistic map f(x) = 4 x (1 - x) with x a function of the Ising couplings. Remarkably, the line bounding the region of chaotic behaviour in x is precisely that defining the Yang-Lee edge singularity in the Ising model. In this paper we show that the one dimensional q-state Potts model for q greater than or equal to 1 also displays such behaviour. A suitable combination of Potts couplings can again be used to define an x satisfying f(x) = 4 x (1 -x). The Yang-Lee…
I discuss the validity of a result put forward recently by Chomaz and Gulminelli [Physica A 330 (2003) 451] concerning the equivalence of two definitions of first-order phase transitions. I show that distributions of zeros of the partition function fulfilling the conditions of the Yang-Lee Theorem are not necessarily associated with nonconcave microcanonical entropy functions or, equivalently, with canonical distributions of the mean energy having a bimodal shape, as claimed by Chomaz and Gulminelli. In fact, such distributions of zeros can also be associated with concave entropy functions and…
To analyze phase transitions in a nonequilibrium system we study its grand canonical partition function as a function of complex fugacity. Real and positive roots of the partition function mark phase transitions. This behavior, first found by Yang and Lee under general conditions for equilibrium systems, can also be applied to nonequilibrium phase transitions. We consider a one-dimensional diffusion model with periodic boundary conditions. Depending on the diffusion rates, we find real and positive roots and can distinguish two regions of analyticity, which can identified with two different ph…
We carry out a numerical and analytic analysis of the Yang-Lee zeros of the 1D Blume-Capel model with periodic boundary conditions and its generalization on Feynman diagrams for which we include sums over all connected and non-connected rings for a given number of spins. In both cases, for a specific range of the parameters, the zeros originally on the unit circle are shown to departure from it as we increase the temperature beyond some limit. The curve of zeros can bifurcate and become two disjoint arcs as in the 2D case. We also show that in the thermodynamic limit the zeros of both Blume-Ca…
Two-time correlation functions of a system of Bose particles are studied. We find relation of zeros of the correlation functions with the Lee-Yang zeros of partition function of the system. Obtained relation gives the possibility to observe the Lee-Yang zeros experimentally. A particular case of Bose particles on two levels is examined and zeros of two-time correlation functions and Lee-Yang zeros of partition function of the system are analyzed.
The phase diagram of the two- and three-state Potts model with infinite-range interactions, in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three state model are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and…
A generalization of the Yang-Lee and Fisher zeros on far-from-equilibrium systems coupled with two thermal baths is proposed. The Yang-Lee zeros were obtained for minimal models which exhibit complicated behavior in the context of the partition function zeros and provide an analitycal treatment. This type of models may be considered as a simplest one and analogous to Ising model for equilibrium. The obtained distributions of generalized Yang-Lee zeros show nontrivial behavior for these simple models.
The microcanonical transfer matrix and its extensions offer a new way of obtaining exact partition functions on finite two dimensional lattices. We show the density of the partition function zeros in the complex x- plane for the Ising model at $y=y_c$ and $y=0.5y_c$, and the density of the partition function zeros in the complex y-plane for the three-state Potts model.
To understand the distribution of the Yang-Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the two-dimensional Yang-Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature beta, can be computed in terms of the Thermodynamics Bethe Ansatz based on its exact S-matrix. We extract the Yang-Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are…
At the Yang-Lee edge singularity, finite-size scaling behavior is used to measure the low-lying excitation spectrum of the Ising quantum spin chain for free boundary conditions. The measured spectrum is used to identify the CFT that describes the Yang-Lee edge singularity of the 2D Ising model for free boundary conditions.
Phase transitions are typically accompanied by non-analytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real-axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros f…
As a foundation of statistical physics, Lee and Yang in 1952 proved that the partition functions of thermal systems can be zero at certain points (called Lee-Yang zeros) on the complex plane of temperature. In the thermodynamic limit, the Lee-Yang zeros approach to real numbers at the critical temperature. However, the imaginary Lee-Yang zeros have not been regarded as experimentally observable since they occur at imaginary field or temperature, which are unphysical. Here we show that the coherence of a probe spin weakly coupled to a many-body system presents zeros as a function of time that a…
We calculate the Yang-Lee zeros for characteristic temperatures of the helix-coil transition in a continuum model of polyalanine. The distribution of these zeros differs from predictions of the Zimm-Bragg theory and supports recent claims that polyalanine exhibits a true phase transition. New estimates for critical exponents are presented and the relation of our results to the Lee-Yang theorem is discussed.
The two-time correlation function for probe spin interacting with spin system (bath) is studied. We show that zeros of this function correspond to zeros of partition function of spin system in complex magnetic field. The obtained relation gives new possibility to observe the Lee-Yang zeros experimentally. Namely, we show that measuring of the time dependence of correlation function allows direct experimental observation of the Lee-Yang zeros.
Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros --- complex singularities of the free energy in systems of finite size --- have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize non-equilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimental…
The complex zeros of partition functions were originally investigated by Lee and Yang to explain the behavior of condensing gases. Since then, Lee-Yang zeros have become a powerful tool to describe phase transitions in interacting systems. Today, Lee-Yang zeros are no longer just a theoretical concept; they have been determined in recent experiments. In one approach, the Lee-Yang zeros are extracted from the high cumulants of thermodynamic observables at finite size. Here, we employ this method to investigate a phase transition in a molecular zipper. From the energy fluctuations in small zippe…
This paper studies the Yang-Lee singularity of the 2-dimensional Ising model on the cylinder via transfer matrix and finite-size scaling techniques. These techniques enable a measurement of the 2-point and 3-point correlations and a comparison of a measurement of a corresponding universal amplitude with a prediction for the amplitude from the (A4,A1) minimal conformal field theory.
Lee-Yang zeros are points on the complex plane of magnetic field where the partition function of a spin system is zero and therefore the free energy diverges. Lee-Yang zeros and their generalizations are ubiquitous in many-body systems and they fully characterize the analytic properties of the free energy and hence thermodynamics of the systems. Determining the Lee-Yang zeros is not only fundamentally important for conceptual completeness of thermodynamics and statistical physics but also technically useful for studying many-body systems. However, Lee-Yang zeros have never been observed in exp…
The Yang-Lee edge singularity is a quintessential nonunitary critical phenomenon accompanied by anomalous scaling laws. However, an imaginary magnetic field involved in this critical phenomenon makes its physical implementation difficult. By invoking the quantum-classical correspondence to embed the Yang-Lee edge singularity in a quantum system with an ancilla qubit, we demonstrate a physical realization of the nonunitary quantum criticality in an open quantum system. Here the nonunitary criticality is identified with the singularity at an exceptional point caused by postselection of quantum m…
Yang-Lee edge singularities are the branch point of the free energy on the complex plane of physical parameters and were shown to be the simplest universality class of phase transitions. However, the Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters, which are unphysical. A recent discovery about the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters. However, how to extract the critical point and the critical exponent of Y…
The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one- and multi-component mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one- and multi-particle reduced density matrices in terms of cluster functions which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang …
We determine the universal location of the Yang-Lee edge singularity in the entire relevant domain of spatial dimensions $1\le d \le 4$ for the Ising universality class. To that end, we present analytical results for $d=1,2,4$ and near four dimensions. For $d=3$ and a set of fractional dimensions, we perform numerical calculations using a systematic Functional Renormalization Group approach.
We discuss the relevance of the Lee-Yang edge singularity to the finite-temperature Z_2-symmetry restoration transition of the Gross-Neveu model in three dimensions. We present an explicit result for its large-N free energy density in terms of Zeta(3) and the absolute maximum of Clausen's function.
We investigate the critical properties of the Lee-Yang model in less than six spacetime dimensions using truncations of the functional renormalization group flow. We give estimates for the critical exponents, study the dependence on the regularization scheme, and show the convergence of our results for increasing size of the truncations in four and five dimensions. While with our truncations it is numerically challenging to approach the three-dimensional case, we provide a simple approximation which allows us to qualitatively study the Lee-Yang model in two and three dimensions, and use it to …
We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. The characterization extends previous results of J. Borcea and the author to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.
We investigate the Yang-Lee edge singularity on non-planar random graphs, which we consider as the Feynman Diagrams of various d=0 field theories, in order to determine the value of the edge exponent. We consider the hard dimer model on phi3 and phi4 random graphs to test the universality of the exponent with respect to coordination number, and the Ising model in an external field to test its temperature independence. The results here for generic (``thin'') random graphs provide an interesting counterpoint to the discussion by Staudacher of these models on planar random graphs.
We study the diluted Ising ferromagnet on the Bethe lattice as a case study for the application of the cavity method to problems with Griffiths-McCoy singularities. Specifically, we are able to make much progress at infinite coupling where we compute, from the cavity method, the density of Lee-Yang zeroes in the paramagnetic Griffiths region as well as the properties of the phase transition to the ferromagnet. This phase transition is itself of a Griffiths-McCoy character albeit with a power law distribution of cluster sizes.
We describe classical stochastic processes by using dynamical Lee-Yang zeros. The system is in contact with external leads and the time evolution is described by the two-state classical master equation. The cumulant generating function is written in a factorized form and the current distribution of the system is characterized by the dynamical Lee-Yang zeros. We show that a continuous distribution of zeros is obtained by discretizing the time variable. When the transition probability is a periodically-oscillating function of time, the distribution of zeros splits into many parts. We study the g…
The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle …
We use the method of `coupling to 2d QG' to study the analytic properties of the universal specific free energy of the O(n) loop model in complex magnetic field. We compute the specific free energy on a dynamical lattice using the correspondence with a matrix model. The free energy has a pair of Yang-Lee edges on the high-temperature sheet and a Langer type branch cut on the low-temperature sheet. Our result confirms a conjecture by A. and Al. Zamolodchikov about the decay rate of the metastable vacuum in presence of Liouville gravity and gives strong evidence about the existence of a weakly m…
The Ising model on annealed complex networks with degree distribution decaying algebraically as $p(K)\sim K^{-λ}$ has a second-order phase transition at finite temperature if $λ> 3$. In the absence of space dimensionality, $λ$ controls the transition strength; mean-field theory applies for $λ>5$ but critical exponents are $λ$-dependent if $λ< 5$. Here we show that, as for regular lattices, the celebrated Lee-Yang circle theorem is obeyed for the former case. However, unlike on regular lattices where it is independent of dimensionality, the circle theorem fails on complex networks when $λ< 5$. …
The form factors of the descendant operators in the massive Lee-Yang model are determined up to level 7. This is first done by exploiting the conserved quantities of the integrable theory to generate the solutions for the descendants starting from the lowest non-trivial solutions in each operator family. We then show that the operator space generated in this way, which is isomorphic to the conformal one, coincides, level by level, with that implied by the $S$-matrix through the form factor bootstrap. The solutions we determine satisfy asymptotic conditions carrying the information about the le…
